Sets and function-of-sets sets have object types {theory of types, logic} {type theory}.
purpose
Distinguishing between types avoids set-theory paradoxes.
types
Sets about objects have type 0. Sets about functions of type-0 sets have type 1. Sets about type-1 sets have type 2. Type-n sets are sets about type n-1 sets.
reducibility
For any type, an equivalent type-0 propositional function exists {axiom of reducibility, type theory} {reducibility axiom, type theory}. Equivalent type-0 propositional functions {relation, type} have classes as members. Classes have object sets. For example, functions can have two variables, and its class can have variable pairs as members.
class
Classes are similar if they have one-to-one correspondence. They are then reflexive, symmetric, and transitive.
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Date Modified: 2022.0224